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On 29 Oct 2012, at 14:36, Stephen P. King wrote:

[Bruno Marchal wrote:] So numbers are universal and can be treated
mathematically as always.

I agree, but the concept of numbers has no meaning prior to the
existence of objects that can be counted. To think otherwise is
equivalent to claiming that unspecified statements are true or false
even in the absence of the possibility of discovering the fact.

Dear Bruno

I think you confuse numbers, and the concept of numbers.

No, I do not. My claim is that Numbers are objects in the mind of
conscious beings. If there does not exist worlds where entities to whom
numbers are concepts then there is no such thing as a concept of numbers
in such worlds. My argument is that concepts of truth and provability of
theorems apply only to the concepts of numbers and their constructions,
not to numbers themselves.

And then your argument is not valid, as with numbers, the miracle is
that we can specify the concept of numbers, as this result in defining
some arithmetical sigma_1 complete theory in terms of 0, s(0), ... and
the laws of addition and multiplication, that everybody understands
(unless philosophers?).

I am a philosopher! My argument rests only on the fact that the
'miracle' is exactly as you state it here: we exist and have a concept
of numbers and can ascertain the truth of arithmetic statements. My
claim is that truth valuations supervene on the ability of consciousness
to form concepts of numbers. I question the entire idea of numbers
existing as separate Platonic entities. In the absence of consciousness,
there is no such thing as a concept!

Bruno

PS BTW, from a computer scientist perspective, your use of NP never
succeed to make sense. I don't dare to ask you to elaborate, as I am
afraid you might aggravate your case. The NP question is fundamental
and has many interesting feature, but it concerns a local tractability
issue, and is a priori, unless justification, not relevant for the
arithmetical body issue, nor number's theology (including physics)
issue, etc.

"There is a close relation between propositions and possible
worlds. We note that every proposition is either true or false at any
given possible world; then the modal status of a proposition is
understood in terms of the worlds in which it is true and worlds in
which it is false."

Solutions to equations or computations are not available until
after they are actually solved. My solution to this is to not go so far
as you do in Step 8. Let me try to be more explicit:

"Instead of linking [the pain I feel] at space-time (x,t) to [a machine
state] at space-time
(x,t), we are obliged to associate [the pain I feel at space-time
(x,t)] to a type or a sheaf of
computations (existing forever in the arithmetical Platonia which
is accepted as existing

independently of our selves with arithmetical realism). "

I am pointing out that the idea of computations "existing
independently of our selves" is wrong in that it conflates *the meaning
and truth valuation of numbers* with *t**he existence of numbers as
Platonic objects*. It is absurd to refer to the claim that the truth of
"17 is prime" depends on any one person or entity, but the claim that
the truth of "17 is prime" is knowable by any person is not absurd. If
we stipulate that the content of knowledge exists somehow prior to that
which knowledge supervenes upon, we are being absurd. The content of
knowledge and the ability of knowledge occur simultaneously or not at all.
Absent the "concept" of numbers there is no such thing as
valuations of numbers because the notion of Platonic objects considers
objects as existing independently as some singular "perfect" version
that is then plurally projected somehow into the physical realm, as we
see in the Allegory of the Cave. This is a one-to-many mapping, not a
one-to-one mapping.
How exactly is a "type" or "sheaf" a singular and "perfect" version
of each and every computation and yet be something that has individuated
valuations? Individual valuations of computations are only those that
occur as physical instantiations of computations and thus they do not
"exist" in Platonia. The Many exist in the physical worlds, no?
I propose a rephrasing of your statement above: We identify the 1p
qualia to a sheaf of computations (as bisimilar Boolean Algebras) that
is dual to physical machine states at diffeomorphically equivalent
space-time coordinates (x, y, z, t). This is a restatement of the Stone
duality into COMP-like terms. ;-) (The idea of diffeomorphic equivalence
is discussed in detail here:
http://plato.stanford.edu/entries/spacetime-holearg/Leibniz_Equivalence.html)

Yes, this is the Pre-Established Harmony, but as I have argued
before this concept is deeply flawed because it tries to claim that
the solution to NP-Hard problem (of choosing the best possible world)
is somehow accessible (for the creation of the monads by God) prior
to the availability of resources with which to actually perform the
computation of the solution. One cannot know the content of a
solution before one computes it, even if one is omniscient!

>>
I don't find any sense.

How is this so difficult for you to comprehend? The Platonic Realm
is defined as timeless, everything in it just 'exists', no? Therefore
any argument that shows that "if A does not exist then neither does B if
B requires A to exist" is true in Platonia as well, (we stipulate the
existence of Platonia as defined
<http://www.wku.edu/%7Ejan.garrett/302/platintr.htm#truebeing> for the
sake of this statement). If a solution to a computation cannot exist
until the computation is run then if the resources required to run the
computation do not exist then there does not exist a solution to the
computation!
I propose that we can easily resolve this conundrum by stating
Computational universality as: "/A computation is universal if and only
if it is independent of any particular physical implementation/." This
allows for the existence of physical implementations, even those that
are themselves defined by correlations between sheaves for computations.
This sets up a relation between computations - as abstract or immaterial
objects - and physical systems that seems consistent with "COMP minus
Step 8". We can recover the picture of step 8,

bijection

in a way that is truly neutral ontologically, by changing its single
directed arrow to a pair of oppositely directed arrows, but this one
that occurs only in the ultimate sense of the elaboration of all
possible physical worlds consistent with Pratt's idea.

This idea, BTW, is consistent with the concept of Indra's Net, as
an inversion of the idea that every Jewel reflects all others: Every
jewel is a physical world that is defined by all computations of it.
Note also that this naturally includes self-computation as jewels also
reflect themselves. ;-)

I hope you don't mind my frankness. I wouldn't say this if I did not
respect some intuition of yours. But math and formalism can't be a
pretext for not doing the elementary reasoning in the philosophy of
mind. If you use math, you have to be clearer on the link with
philosophy or theology. To be understandable by others.

I am trying to be clear. I will correct and rephrase my verbiage
until you understand it. I reject the idea of an entity, 'God', whose
total purpose is to "observe" the Reality of the Universe! If we accept
the idea that numbers exist in our complete absence, then it follows
that an entity like us cannot exist just to observe the existence of
numbers (or anything else). Why postulate the existence of a special
entity that does what we collectively are already doing?
It is our collective consciousness that Constitutes the Platonic
Realm, IMHO. A theory that there is some independently existing realm is
a gross violation of Occam.

--
Onward!
Stephen
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